The problem asks us to find the value of $a$ in the quadratic equation $2x² - 3x + a = 0$, given that its roots are in the ratio $1 : 2$.

Using Vieta's Formulas

Let the roots of the quadratic equation $Ax² + Bx + C = 0$ be $p$ and $q$. According to Vieta's formulas:

• Sum of roots: $p + q = -B/A$
• Product of roots: $p * q = C/A$

In our equation, $2x² - 3x + a = 0$, we have $A = 2$, $B = -3$, and $C = a$.

Applying the Ratio of Roots

We are given that the roots are in the ratio $1 : 2$. Let the roots be $k$ and $2k$.

Sum of roots:

$k + 2k = -(-3)/2$

$3k = 3/2$

Solving for $k$:

$k = (3/2) / 3$

$k = 1/2$

Product of roots:

$k * 2k = a/2$

$2k² = a/2$

Calculating the Value of 'a'

Now, substitute the value of $k = 1/2$ into the product of roots equation:

$2 * (1/2)² = a/2$

$2 * (1/4) = a/2$

$1/2 = a/2$

Multiply both sides by 2:

$a = 1$

Therefore, the value of $a$ is 1.